Finite Group Databasebeta

← #21427423

Subgroups — #21427423

Order 7552.

1116 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 1116 conjugacy classes of subgroups — too many to draw. Only its 181 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C22⋊D59C211yesmaximal, characteristic
2C2×#14273710C212yesmaximal, complemented
3#16837317C214yesmaximal, complemented
4#16837281C212yesmaximal, complemented
5#16837274C212yesmaximal, complemented
6#16837258C211yesmaximal, characteristic, complemented
7#16837257C212yesmaximal, complemented
8C22×C22⋊C4×C59C211yesmaximal, characteristic, complemented
9C24×D59C2211yescharacteristic
10C22×C22⋊D59—24no—
11C22×C22⋊D59—28no—
12C22×C22⋊D59C2212yes—
13C22×C22⋊D59C2212yes—
14C23×Dic59C2211yescharacteristic
15#14273710—28no—
16C2×Dic59⋊C4C2214yescomplemented
17C2×C22⋊Dic59C2212yes—
18C2×C22⋊Dic59C2212yescomplemented
19C2×C22⋊Dic59C2212yescomplemented
20C2×(C2×D59)⋊C4C2214yescomplemented
21C2×(C2×D59)⋊C4C2214yescomplemented
22C59⋊#95C2214yescomplemented
23C25×C59C2211yescharacteristic
24C23×C4×C59C2211yescharacteristic, complemented
25C23×C4×C59C2211yescharacteristic, complemented
26C2×C22⋊C4×C59—24no—
27C2×C22⋊C4×C59C2212yes—
28C2×C22⋊C4×C59C2212yescomplemented
29C24×C59—24no—
30C24×C59—24no—
31C24×C59—42no—
32C24×C59D414yes—
33C24×C59C2311yescharacteristic
34C24×C59C2311yescharacteristic
35C24×C59C2311yescharacteristic
36C23×D59—22no—
37C23×D59—24no—
38C23×D59—24no—
39C23×D59—42no—
40C23×D59C2312yes—
41C22×C4×C59—22no—
42C22×C4×C59—22no—
43C22×C4×C59—24no—
44C22×C4×C59—42no—
45C22×C4×C59C2312yes—
46C22×C4×C59C2312yes—
47C2×C22⋊D59—232no—
48C2×C22⋊D59—24no—
49C2×C22⋊D59—24no—
50C2×C22⋊D59—24no—
51C2×C22⋊D59—24no—
52C2×C22⋊D59—24no—
53C2×C22⋊D59D418yes—
54C22×Dic59—22no—
55C22×Dic59—24no—
56C22×Dic59C2314yes—
57C22×Dic59C2312yes—
58C22⋊C4×C59—24no—
59C22⋊C4×C59—24no—
60C22⋊Dic59—24no—
61C22⋊Dic59—24no—
62C22⋊Dic59—42no—
63(C2×D59)⋊C4—28no—
64(C2×D59)⋊C4—44no—
65Dic59⋊C4—28no—
66C23×C59—22no—
67C23×C59—22no—
68C23×C59—22no—
69C23×C59—22no—
70C23×C59—28no—
71C23×C59—28no—
72C23×C59—22no—
73C23×C59—22no—
74C23×C59—22no—
75C23×C59—44no—
76C23×C59—416no—
77C23×C59C2×D414yes—
78C23×C59C2×D412yes—
79C23×C59C2×D412yes—
80C23×C59C2×D412yes—
81C23×C59#4114yes—
82C23×C59C2411yescharacteristic
83C22×D59—28no—
84C22×D59—22no—
85C22×D59—22no—
86C22×D59—24no—
87C22×D59—24no—
88C22×D59—416no—
89C22×D59—44no—
90C22×D59—44no—
91C22×D59C2×D414yes—
92C2×C4×C59—24no—
93C2×C4×C59—22no—
94C2×C4×C59—22no—
95C2×C4×C59—24no—
96C2×C4×C59—44no—
97C2×C4×C59—44no—
98C22⋊D59—216no—
99C22⋊D59—216no—
100C2×Dic59—28no—
101C2×Dic59—22no—
102C2×Dic59—22no—
103C2×Dic59—24no—
104C2×Dic59—24no—
105C2×Dic59—24no—
106C2×Dic59C2×D414yes—
107C22×C59—22no—
108C22×C59—22no—
109C22×C59—22no—
110C22×C59—22no—
111C22×C59—28no—
112C22×C59—28no—
113C22×C59—22no—
114C22×C59—22no—
115C22×C59—22no—
116C22×C59—416no—
117C22×C59—44no—
118C22×C59#12014yes—
119C22×C59#12314yescomplemented
120C22×C59C22×D412yes—
121C22×C59C22×D411yescharacteristic
122C22×C59C2×#4112yes—
123C22×C59H(5,𝔽2)12yes—
124C2×D59—24no—
125C2×D59—28no—
126C2×D59—416no—
127C2×D59—42no—
128C2×D59—42no—
129C2×D59—44no—
130C4×C59—42no—
131C4×C59—42no—
132Dic59—24no—
133Dic59—44no—
134#2405—591nomaximal, Sylow
135C2×C59—24no—
136C2×C59—24no—
137C2×C59—42no—
138C2×C59C2×#12011yescharacteristic
139C2×C59C2×#12311yescharacteristic
140C2×C59#53411yescharacteristic
141C2×C59#54614yes—
142D59—44no—
143C23×D4—591no—
144C2×#123—592no—
145C22×C22⋊C4D5911yescharacteristic, complemented
146#394—594no—
147#386—592no—
148#385—592no—
149#379—591no—
150#379—592no—
151C59#240511yescharacteristic, complemented, Sylow
152C25—591no—
153C25C2×D5911yescharacteristic
154C22×D4—592no—
155C22×D4—592no—
156C22×D4—1188no—
157C22×D4—1184no—
158C23×C4—591no—
159C23×C4C2×D5911yescharacteristic, complemented
160C23×C4C2×D5911yescharacteristic, complemented
161#123—1188no—
162C2×C4⋊C4—594no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—592no—
165C2×C22⋊C4—592no—
166C2×C22⋊C4—594no—
167C2×C22⋊C4—592no—
168C2×C22⋊C4—594no—
169C2×C22⋊C4C2×D5912yes—
170C2×C22⋊C4C2×D5912yescomplemented
171#95—594no—
172C24—24no—
173C24—24no—
174C24—42no—
175C24—592no—
176C24—1184no—
177C24—1182no—
178C24—1184no—
179C24—2362no—
180C24C22⋊D5914yes—
181C24C22×D5911yescharacteristic
182C24C22×D5911yescharacteristic
183C24C22×D5911yescharacteristic
184C2×D4—598no—
185C2×D4—1184no—
186C2×D4—1184no—
187C2×D4—11832no—
188C2×D4—1184no—
189C2×D4—1184no—
190C2×D4—1184no—
191C22×C4—24no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—42no—
195C22×C4—592no—
196C22×C4—594no—
197C22×C4—1184no—
198C22×C4—1182no—
199C22×C4C22×D5912yes—
200C22×C4C22×D5912yes—
201C4⋊C4—1188no—
202C22⋊C4—24no—
203C22⋊C4—24no—
204C22⋊C4—1188no—
205C22⋊C4—1184no—
206C22⋊C4—1184no—
207C22⋊C4—2364no—
208C22⋊C4—2362no—
209C23—22no—
210C23—22no—
211C23—22no—
212C23—22no—
213C23—28no—
214C23—28no—
215C23—22no—
216C23—22no—
217C23—22no—
218C23—44no—
219C23—416no—
220C23—594no—
221C23—1188no—
222C23—1182no—
223C23—1182no—
224C23—1184no—
225C23—1184no—
226C23—23616no—
227C23—2364no—
228C23—2364no—
229C23#1174568514yes—
230C23D4×D5914yes—
231C23C2×C22⋊D5912yes—
232C23C2×C22⋊D5912yes—
233C23C2×C22⋊D5912yes—
234C23C23×D5911yescharacteristic
235D4—11816no—
236D4—11816no—
237C2×C4—24no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—24no—
241C2×C4—44no—
242C2×C4—44no—
243C2×C4—594no—
244C2×C4—1188no—
245C2×C4—1182no—
246C2×C4—1182no—
247C2×C4—1184no—
248C2×C4—1184no—
249C2×C4—1184no—
250C22—22no—
251C22—22no—
252C22—22no—
253C22—28no—
254C22—28no—
255C22—22no—
256C22—22no—
257C22—22no—
258C22—22no—
259C22—416no—
260C22—44no—
261C22—1188no—
262C22—1184no—
263C22—23616no—
264C22—2362no—
265C22—2362no—
266C22—2364no—
267C22#1427369514yes—
268C22#1427371014yescomplemented
269C22C2×D4×D5912yes—
270C22C22×C22⋊D5911yescharacteristic
271C22C2×#1174568512yes—
272C22#1427377212yes—
273C4—42no—
274C4—42no—
275C4—1184no—
276C4—2364no—
277C2—24no—
278C2—24no—
279C2—42no—
280C2—2364no—
281C2C2×#1427369511yescharacteristic
282C2C2×#1427371011yescharacteristic
283C2#1683796111yescharacteristic
284C2#1683804214yes—
subgroupG/H
center, FrattiniC23C23×D59
commutator, socleC23×C59C24
FittingC22×C22⋊C4×C59C2
hypercenterC22×C22⋊C4D59
hypocenterC59#2405
LCSG ▷ C23×C59 ▷ C59
UCSC1 ◁ C23 ◁ C22×C22⋊C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#14273710C2semidirect
3. #16837317C2semidirect
4. #16837281C2semidirect
5. #16837274C2semidirect
6. #16837258C2semidirect
7. #16837257C2semidirect
8. C22×C22⋊C4×C59C2semidirect
16. C2×Dic59⋊C4C22semidirect
18. C2×C22⋊Dic59C22semidirect
19. C2×C22⋊Dic59C22semidirect
20. C2×(C2×D59)⋊C4C22semidirect
21. C2×(C2×D59)⋊C4C22semidirect
22. C59⋊#95C22semidirect
24. C23×C4×C59C22semidirect
25. C23×C4×C59C22semidirect
28. C2×C22⋊C4×C59C22semidirect
119. C22×C59#123semidirect
145. C22×C22⋊C4D59semidirect
151. C59#2405semidirect
159. C23×C4C2×D59semidirect
160. C23×C4C2×D59semidirect
170. C2×C22⋊C4C2×D59semidirect
268. C22#14273710semidirect